Yes, it is possible, and has been done.
The GML Compilation
I would say the handiest resource available for this online is the "Mythical Chronology Chart" page on Carlos Parada's website the Greek Mythology Link (GML).
As further explained on the afore-linked page (q.v.),
The chart does not establish precise historical dates, but shows the
relative position in time of certain legends and characters as may be
deduced from the legends themselves...
Each person in the lists of mortals counts as one generation, three
generations completing one century. Continuity has been established,
in most cases, through parentage or throne succession. Mortals written
in red fought in the Trojan War. It may be seen, by comparing the
lists horizontally, who were living at the same time.
Further down the page there is drawn up a quite neat "List of Events" starting with "Creation of the World", after which the first event with a date listed is the life of one of the earliest kings of Argos in the 1870s BC, going from thence down to the reigns, in 550 BC, of Tarquinius Superbus in Rome, and of Croesus over Lydia.
At the bottom of the page, events get fleshed out in a little more detail comparing, with one another, dates as broken down by three different sources: the Marmor Parium (264 BC), Denis Pétau (1628 AD) and Giambattista Vico (1744 AD).
So for example, according to Vico, the Titan Prometheus steals fire from the gods in 2143 BC while the Romans drive the Tarquin family off the throne of their city in 551 or 561 BC.
There a few other places online providing similar information with the same general chronological structure but in bare list formats, with varying conjectures for dates of when the different events would have taken place:
Marc Carlson's 2004 "Timeline of Classical
explains that he uses the traditional foundation-year of Rome, 753 BC, as
the linchpin of his dates, counting everything backwards from there.
(He does list a few historical occurrences after that event.)
Argyros George Argyrou's 2009 "Chronology of Greece and the Greek
Myths" starts off
with the foundation of Atlantis and ends, in 30 BC, with the death
of Cleopatra and of Mark Antony.
HellenicaWorld consolidates information derived from the GML, from
Argyrou, and from Wikipedia, into its own "Mythical Chronology of
Greece", regarding which it says that:
This list largely reflects the work of Saint Jerome, whose work in
turn was based primarily on the analysis of Apollodorus, Diodorus
Siculus, and Eusebius... In a few cases, the chronology also reflects
the opinions of more recent scholars, who have cross referenced the
mythology to archeological discoveries.
Thanks to the fact that something like at least 95% of all the characters of Greek mythology are traceably related to one another, with most of them being either divine or royalty (or both), it is a fairly straightforward task to locate virtually any one of them within a particular city's king-list or within the lifetime of at least one other character, who will then inadvertently be connected to yet other personages.
Oedipus lived about a generation and a half before the Trojan War. We know this because the biggest conflict involving Greek mortals before the Trojan War was the expedition of the Seven Against Thebes. The cause of this contest was a quarrel between Eteocles and Polyneices over the throne of their father Oedipus. The young men who fought in the war of the Epigoni, ten years after that of the Seven Against Thebes, were among the Greek heroes of the Trojan War, which took place some years after the Epigoni had sacked Thebes. One of these Epigoni who went to Troy was Oedipus's grandson Thersander.
We can also deduce that the Argonautic expedition took place at some point during Oedipus' reign over Thebes because Heracles [Hercules] was Argonaut. Heracles was at some point married to Megara, a daughter of Creon, who had been regent king of Thebes just before Oedipus acceded to the throne. Moreover, after the attack of the Seven Against Thebes, Heracles had placed Laodamas, yet another grandson of Oedipus, on the throne of Thebes.
See the GML's chart above for how Oedipus, Polyneices, and Thersander (the last of these being Polyneices' son, and Laodamas' cousin), fit into the overall royal Theban succession.
The Wars of the Titans and the Giants
The human race did not yet exist at the time of the Titanomachy, unless you count the version of the myth of the Five "Ages" of Man, in which the first race or breed of men was made of gold and lived during the reign of Cronus. Still, humans are not said to have participated in this war, and they are otherwise more commonly regarded as having been created by the Olympians, in a collaboration project with the Titan Prometheus, a good while after the end of the contest between Cronus and Zeus.
The Gigantomachy took place during the lifetime of Heracles, thus about a generation before the Trojan War. Zeus, who, one generation earlier, had foreseen this war, brought Heracles to life specifically in order for him to aid the Olympians in this conflict, for it had been prophesied that the gods would be defeated unless there was a mortal who could assist them.
The Roman writer Hyginus claims that, according to the playwright Aeschylus, Zeus had imprisoned Prometheus on Mt Caucasus for thirty thousand years before Heracles freed the Titan. Using this time-frame it would mean that, since Prometheus was imprisoned only after the end of the Titanomachy, the war between Cronus and Zeus took place thirty millennia before the Giants' uprising against the Olympians. Of the resources listed above, only Carlson's Timeline refelcts this, placing the start of Prometheus' imprisonment at c. 31,270 BC.
Pausanias tantalisingly mentions that Heracles once fought against a giant named Thurius, while Tyndareus fought against a certain Eurytus, but then he refuses to narrate these events. There are several characters in Greek myth named Eurytus, and one of them is among the Giants of the Gigantomachy. If this is supposed to be the same as the one against whom Tyndareus fought, then Tyndareus would be the only other mortal, apart from Heracles, who is mentioned in connection with the Giants' War.
Granted, they could have engaged in combat before the Giants attacked the gods. There are, however, several other contemporaries of Tyndareus bearing this name who would serve as more plausible opponents for him, going by the other stories and adventures involving this particular hero. (It could just as easily be the Centaur Eurytus intended here by Pausanias.)
The Births of Hephaestus and Athena
The gods, particularly the Olympian ones, had dramatically brief childhoods. Surprisingly Zeus seems to be the exception here, who may have had a more ordinary human-length infancy and childhood. Conversely Zeus's children are described as having been born either full-grown (like Athena) or to have grown up in a matter of hours or just a few days (like Artemis and Apollo, and perhaps Hermes).
With this in mind I don't see the slightly different versions of Athena's birth presenting any difficulty for the timeline. Whoever happens to be older between Hephaestus and Athena, the likelihood is that their age difference is negligibly small either way (even if it's as big as a few years rather than just days).
Besides that, either version fits in neatly enough with the rest of the chronology, plus we have the alternative in which, if Hephaestus, for whatever reason, is an unavailable surgeon at Athena's birth, Apollodorus tells us that it was Prometheus instead who broke Zeus's head open in order to grant Athena her birth.
Whatever the case, by my guesstimate, all of the original Twelve Great Olympians—with the exception of Hermes—were born before the end of the Titanomachy, and so whichever combination of the drama involved in the births of Hephaestus and Athena would have happened within the nine years of the war, or before it.
How many generations Greek mythology spans would depend on how you define the end-point of the mythology. It blends fairly seamlessly into Roman mythology, wherein the only especially compelling deviation is the language and names of the characters, which become Latin, and Latinised. Nonetheless a chunk of the mythography regarding the foundation of Rome and the events leading up to it after the Trojan War comes from Greek sources writing in Roman times, still giving a very Greek perspective on what they are narrating. Dionysius of Halicarnassus, for example, is at great pains to emphasise the Greek origins of the Roman state and its peoples. Meanwhile Servius, commenting on Virgil's Aeneid, insists that the ancestors of the Dardanians and Trojans, who then in turn gave rise to the Romans, had themselves originally come from Italy rather than Greece.
Beyond that there is the fact that the further into historical times that mythology encroaches, the more it blends in, transforming into "legend" and "folklore." So for example, there are numerous tall tales about Alexander the Great (356 - 323 BC) which together make up a collection of fables known as the Alexander Romance. Included in the legends about him is the idea, apparently promoted even while he was still alive, that he was a son of Zeus. This technically makes him a character within what we call Greek mythology, and he is treated as such, e.g., in the Recognitions which are ascribed to Clement of Alexandria, where he is listed among other well-known children of Zeus, such as Epaphus, Tantalus and Sarpedon, all of whom, unlike Alexander, would have lived squarely within the "mythical" era.
Then down into Christian times, as late as the 400s AD, there are stories of ordinary people encountering Centaurs, Satyrs and Tritons. Meanwhile in Modern Greek folklore, Nereids, the ancient goddesses of the Mediterranean Sea, together with other, more minor mythical creatures, have survived as supernatural creatures in much the same way that fairies are the old legendary Celtic phenomenon occupying the popular imagination of the British Isles (and inadvertently having been exported from thence pretty much all over the world).
Often a clear end-point, if one must have such a thing, comes at the conclusion of a particular city's king-list. It's fairly easy to note where the monarchs of a certain state stop being mythical personages and start being historical figures. The lineage of Alexander the Great, for example, places the Macedonian conqueror about fourteen generations away from Karanos [Caranus], the founder of the Argead dynasty and first ruler of the historical kingdom of Macedon, who reigned four centuries before Alexander's time. Karanos is purported to have been a son of Temenus, who in turn was a great-great-grandson of the mighty Heracles himself.
So in this instance one might say that the mythology "ends" with the adventures of Temenus and of his contemporaries in other city-states who are said have been the fathers of the "historical" kings in places such as Messenia, Sparta, Athens and Ionia, c. 800 BC, about half a century before Romulus and Remus, the twin sons of the Italian god Mars, found Rome.
How Many Generations?
Counting up the generations on the GML chart, starting with Phoroneus, whose subjects in Argos claimed that he was the first man, down to Tarquinius Superbus in Rome, gives us roughly forty generations. Note, however, that this does not include the majority of the thirty thousand years which Hyginus gives us between the Titanomachy and the time of Heracles, during which time, according to one interpretation of Hesiod's Works and Days, the earth would have been populated by successive breeds of generally nameless golden, silver and bronze men, prior to the advent of the Age of Heroes.
Also, according to the information provided by Plato's Critias dialogue, the kingdom of Atlantis would have been founded at least seven thousand years earlier than the 2094 BC date which Argyrou's Chronology supplies for it; and Athens is apparently just as old as Atlantis. The first inhabitants of Atlantis were Autochthons (people born from the ground) whose daughter married the sea-god Poseidon; a number of variant origin myths about Athens consistently present all of its first kings as Autochthons also.
Heroes Born Before the Olympians
There are a few stories regarding the childhoods of some of the Olympian deities in which they are reared by mortal heroes, which does indeed mean that there are, in at least some versions of the mythology, heroes who are in fact older than the Olympians.
According to one tradition, Athena was raised in the Boeotian village of Alalcomenia by the man after whom the place was named, a certain Alalcomenes or Alalcomeneus, who was a local Autochthon, whose wife was called Athenaïs and whose son was named Glaucopus, "Grey-Eyed," a feature he apparently shared with the goddess. From her foster father the deity was known as Athena Alalcomenia.
In their two different Metamorphoses epics, Ovid, and Antoninus Liberalis, respectively, contradict what I say above regarding the human race not existing during the Titans' War. A rabble of Lycian peasants is said to have mistreated the Titaness Leto when she was pregnant with her twins Apollo and Artemis, for which she punished these men by turning them into the first frogs. Since I place the birth of these twin deities during the Titanomachy, my explanation is that the peasants, like Alalcomenes and the first Atlanteans, were Autochthons, and thus sort of human, but not the general rendition thereof which came later under the auspices of Prometheus.
During Hermes' adventures of thievery on the same day in which he was born, he turned an old man named Battus to stone after he discovered that he couldn't trust the man not to tell on him for stealing Apollo's cattle.
Hyginus reports the wildly contradictory story which says that a pair of nymphs named Aex and Helice nursed Zeus when the god was a baby. The problem is that the nymphs were the daughters of a certain Olenus who was a son of Hephaestus, who we know was Zeus's own son (or at least the son of Zeus' wife Hera)! How Zeus's own great-granddaughters (or great-great-nieces) are taking care of him in his infancy is not explained.
Alternatively Zeus is said to have been reared by Amaltheia, daughter of Haemonius; or by the daughters of a Cretan king called Melisseus. Theoi.com interprets the names of these mysterious men as belonging to some of the Curetes and Dactyls who are usually ascribed with having looked after the infant god.
Dionysus and Heracles, both of whom became gods, according to Diodorus Siculus, were each granted the title of "Olympian" by their father Zeus. Both of them were born to mortal women somewhere in the middle of the aforementioned forty-generation time-span and are therefore younger than numerous other heroes.